We're given an finite grid of random variables like so: $$ \begin{bmatrix} A & B &... \\C & \ddots \\ \vdots&&X\\ \end{bmatrix} $$
a subset of variables on the gird is independent if no two variables share a row and no two variables share a column.
A very small nontrivial example of a grid with this property could be $$ \begin{bmatrix} z_1 z_a & z_2 z_a \\ z_1 z_b & z_2 z_b\end{bmatrix} $$ with $z_i$ independent identically distributed Gaussians.
If this is perhaps too broad, so consider a case where permutation of rows or columns doesn't change the overall probability distribution. (Example above is one such case)
This started from trying to draw from distribution over such grid, but seems interesting in itself. What can be said about such collections of random variables? How much information do we need to describe the probability distribution of such grids?
